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Theorem qnumdenbi 12194
Description: Two numbers are the canonical representation of a rational iff they are coprime and have the right quotient. (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
qnumdenbi ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (((𝐡 gcd 𝐢) = 1 ∧ 𝐴 = (𝐡 / 𝐢)) ↔ ((numerβ€˜π΄) = 𝐡 ∧ (denomβ€˜π΄) = 𝐢)))

Proof of Theorem qnumdenbi
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 opelxpi 4660 . . . 4 ((𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ⟨𝐡, 𝐢⟩ ∈ (β„€ Γ— β„•))
213adant1 1015 . . 3 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ⟨𝐡, 𝐢⟩ ∈ (β„€ Γ— β„•))
3 qredeu 12099 . . . 4 (𝐴 ∈ β„š β†’ βˆƒ!π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))))
433ad2ant1 1018 . . 3 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ βˆƒ!π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))))
5 fveq2 5517 . . . . . . 7 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ (1st β€˜π‘Ž) = (1st β€˜βŸ¨π΅, 𝐢⟩))
6 fveq2 5517 . . . . . . 7 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ (2nd β€˜π‘Ž) = (2nd β€˜βŸ¨π΅, 𝐢⟩))
75, 6oveq12d 5895 . . . . . 6 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ ((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = ((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)))
87eqeq1d 2186 . . . . 5 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ (((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ↔ ((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = 1))
95, 6oveq12d 5895 . . . . . 6 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)) = ((1st β€˜βŸ¨π΅, 𝐢⟩) / (2nd β€˜βŸ¨π΅, 𝐢⟩)))
109eqeq2d 2189 . . . . 5 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ (𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)) ↔ 𝐴 = ((1st β€˜βŸ¨π΅, 𝐢⟩) / (2nd β€˜βŸ¨π΅, 𝐢⟩))))
118, 10anbi12d 473 . . . 4 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ ((((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))) ↔ (((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = 1 ∧ 𝐴 = ((1st β€˜βŸ¨π΅, 𝐢⟩) / (2nd β€˜βŸ¨π΅, 𝐢⟩)))))
1211riota2 5855 . . 3 ((⟨𝐡, 𝐢⟩ ∈ (β„€ Γ— β„•) ∧ βˆƒ!π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) β†’ ((((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = 1 ∧ 𝐴 = ((1st β€˜βŸ¨π΅, 𝐢⟩) / (2nd β€˜βŸ¨π΅, 𝐢⟩))) ↔ (β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) = ⟨𝐡, 𝐢⟩))
132, 4, 12syl2anc 411 . 2 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ((((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = 1 ∧ 𝐴 = ((1st β€˜βŸ¨π΅, 𝐢⟩) / (2nd β€˜βŸ¨π΅, 𝐢⟩))) ↔ (β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) = ⟨𝐡, 𝐢⟩))
14 op1stg 6153 . . . . . 6 ((𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (1st β€˜βŸ¨π΅, 𝐢⟩) = 𝐡)
15 op2ndg 6154 . . . . . 6 ((𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (2nd β€˜βŸ¨π΅, 𝐢⟩) = 𝐢)
1614, 15oveq12d 5895 . . . . 5 ((𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = (𝐡 gcd 𝐢))
17163adant1 1015 . . . 4 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = (𝐡 gcd 𝐢))
1817eqeq1d 2186 . . 3 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = 1 ↔ (𝐡 gcd 𝐢) = 1))
19143adant1 1015 . . . . 5 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (1st β€˜βŸ¨π΅, 𝐢⟩) = 𝐡)
20153adant1 1015 . . . . 5 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (2nd β€˜βŸ¨π΅, 𝐢⟩) = 𝐢)
2119, 20oveq12d 5895 . . . 4 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ((1st β€˜βŸ¨π΅, 𝐢⟩) / (2nd β€˜βŸ¨π΅, 𝐢⟩)) = (𝐡 / 𝐢))
2221eqeq2d 2189 . . 3 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (𝐴 = ((1st β€˜βŸ¨π΅, 𝐢⟩) / (2nd β€˜βŸ¨π΅, 𝐢⟩)) ↔ 𝐴 = (𝐡 / 𝐢)))
2318, 22anbi12d 473 . 2 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ((((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = 1 ∧ 𝐴 = ((1st β€˜βŸ¨π΅, 𝐢⟩) / (2nd β€˜βŸ¨π΅, 𝐢⟩))) ↔ ((𝐡 gcd 𝐢) = 1 ∧ 𝐴 = (𝐡 / 𝐢))))
24 riotacl 5847 . . . . . . 7 (βˆƒ!π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))) β†’ (β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) ∈ (β„€ Γ— β„•))
25 1st2nd2 6178 . . . . . . 7 ((β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) ∈ (β„€ Γ— β„•) β†’ (β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) = ⟨(1st β€˜(β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))))), (2nd β€˜(β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))))⟩)
263, 24, 253syl 17 . . . . . 6 (𝐴 ∈ β„š β†’ (β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) = ⟨(1st β€˜(β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))))), (2nd β€˜(β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))))⟩)
27 qnumval 12187 . . . . . . 7 (𝐴 ∈ β„š β†’ (numerβ€˜π΄) = (1st β€˜(β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))))))
28 qdenval 12188 . . . . . . 7 (𝐴 ∈ β„š β†’ (denomβ€˜π΄) = (2nd β€˜(β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))))))
2927, 28opeq12d 3788 . . . . . 6 (𝐴 ∈ β„š β†’ ⟨(numerβ€˜π΄), (denomβ€˜π΄)⟩ = ⟨(1st β€˜(β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))))), (2nd β€˜(β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))))⟩)
3026, 29eqtr4d 2213 . . . . 5 (𝐴 ∈ β„š β†’ (β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) = ⟨(numerβ€˜π΄), (denomβ€˜π΄)⟩)
3130eqeq1d 2186 . . . 4 (𝐴 ∈ β„š β†’ ((β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) = ⟨𝐡, 𝐢⟩ ↔ ⟨(numerβ€˜π΄), (denomβ€˜π΄)⟩ = ⟨𝐡, 𝐢⟩))
32313ad2ant1 1018 . . 3 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ((β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) = ⟨𝐡, 𝐢⟩ ↔ ⟨(numerβ€˜π΄), (denomβ€˜π΄)⟩ = ⟨𝐡, 𝐢⟩))
33 qnumcl 12190 . . . . 5 (𝐴 ∈ β„š β†’ (numerβ€˜π΄) ∈ β„€)
34 qdencl 12191 . . . . 5 (𝐴 ∈ β„š β†’ (denomβ€˜π΄) ∈ β„•)
35 opthg 4240 . . . . 5 (((numerβ€˜π΄) ∈ β„€ ∧ (denomβ€˜π΄) ∈ β„•) β†’ (⟨(numerβ€˜π΄), (denomβ€˜π΄)⟩ = ⟨𝐡, 𝐢⟩ ↔ ((numerβ€˜π΄) = 𝐡 ∧ (denomβ€˜π΄) = 𝐢)))
3633, 34, 35syl2anc 411 . . . 4 (𝐴 ∈ β„š β†’ (⟨(numerβ€˜π΄), (denomβ€˜π΄)⟩ = ⟨𝐡, 𝐢⟩ ↔ ((numerβ€˜π΄) = 𝐡 ∧ (denomβ€˜π΄) = 𝐢)))
37363ad2ant1 1018 . . 3 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (⟨(numerβ€˜π΄), (denomβ€˜π΄)⟩ = ⟨𝐡, 𝐢⟩ ↔ ((numerβ€˜π΄) = 𝐡 ∧ (denomβ€˜π΄) = 𝐢)))
3832, 37bitrd 188 . 2 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ((β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) = ⟨𝐡, 𝐢⟩ ↔ ((numerβ€˜π΄) = 𝐡 ∧ (denomβ€˜π΄) = 𝐢)))
3913, 23, 383bitr3d 218 1 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (((𝐡 gcd 𝐢) = 1 ∧ 𝐴 = (𝐡 / 𝐢)) ↔ ((numerβ€˜π΄) = 𝐡 ∧ (denomβ€˜π΄) = 𝐢)))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  βˆƒ!wreu 2457  βŸ¨cop 3597   Γ— cxp 4626  β€˜cfv 5218  β„©crio 5832  (class class class)co 5877  1st c1st 6141  2nd c2nd 6142  1c1 7814   / cdiv 8631  β„•cn 8921  β„€cz 9255  β„šcq 9621   gcd cgcd 11945  numercnumer 12183  denomcdenom 12184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-sup 6985  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-dvds 11797  df-gcd 11946  df-numer 12185  df-denom 12186
This theorem is referenced by:  qnumdencoprm  12195  qeqnumdivden  12196  divnumden  12198  numdensq  12204
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